Just Say "Charge It!"

Just Say "Charge It!"Author(s): Connie H. Yarema and John H. SampsonSource: The Mathematics Teacher, Vol. 94, No. 7 (October 2001), pp. 558-564Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20870797 .

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Connie H. Yarema and John H. Sampson

Credit arge It!"

In 1996, the average

household with credit cards owed more than

$7000 and paid more than $1000 in interest and fees

?common response to a salesclerk's it a purchase will be a cash or credit

.ottgh credit cards did not appear ^ United States until 1915 (Johnson and

Mowry 1998), they are now considered "plastic money" and are frequently used. According to a

report issued in December 1997 by the nonprofit Consumer Federation of America in Washington, D.C., the estimated 55 million to 60 million house holds with revolving credit cards carried an average card debt of more than $7000 per household. The consumer group also estimated that these house holds paid an average of more than $1000 in inter est and fees during 1996.

Because inability to pay debt leads to higher rates and fees for everyone (Feldstein 1998), the increase in the amount of credit-card debt has been brought to national attention. The 16 September 1998

"Fleecing of America" segment on NBC's Nightly News attributed an increase in personal bankruptcy during the 1990s to credit-card debt among college students, especially freshmen, who were suddenly expected to be credit-wise adults. Furthermore, the telecast commented that most of these young adults have no concept of how interest works. The Con sumer Federation of America (1999) issued a report stating that students' credit-card debt is under estimated and is related to such phenomena as sui cides of indebted students (www.consumerfed.org). About 78 percent of undergraduates in the United States have at least one credit card and incur an

average debt of more than $2700 per cardholder (Nellie Mae 2001). In addition, one-tenth of four

year college students owe more than $7000. The Consumer Federation of America report indicates that few credit-card issuers or institutions of higher education support effective financial education of freshmen. However, in a report for National Public Radio on 9 September 1998, Elaine Cory ascribed some of this financial illiteracy among postsecondary students not to institutions of higher education but to secondary schools, stating that they do not do their part in teaching about debt.

Sampson, a coauthor of this article, became aware of this issue through a project of the Louisiana Sys temic Initiative Program (LaSIP) and Rural Systemic

Initiative (RSI) that was codirected and instructed

by Yarema, the other author of this article. Because

Sampson wants his students to be knowledgeable about credit-card debt, he presents the subject in his second-year algebra class as an application of

sequences. The credit-card-debt activity described in this article evolved from an activity first devel

oped by Yarema for her college-level algebra stu dents. The authors believe that teachers who use the

activity while teaching geometric and arithmetic

sequences can increase their students' awareness of

underlying mathematical concepts that are relevant to the world of commerce and personal finance (NCTM 2000, pp. 66, 297).

A recursive approach to studying the mathemat ics of finance is not new. Most recursive applica tions model the growth of an investment or are used to develop general finance formulas. For ex

ample, Curriculum and Evaluation Standards for School Mathematics (1989, pp. 132-34) illustrates the development of a compound-interest investment formula through patterns, sequences, and general izations of patterns. The chart in figure 1 demon strates the process of generalizing the sequence of the accumulated amount of money in an account that is compounded yearly. A0 is the initial deposit,

is the compounding period, r is the annual com

pounding rate, and Ah A2,..., An is the geometric sequence formed from recursively calculating the amount in the account after each compounding period. The recursive approach helps students con

ceptually understand the role that interest plays in the growth of an investment and understand how to look at patterns to develop the general formula for compound interest.

Technological tools allow the applications of

sequences to go beyond investment topics and

Connie Yarema, [email protected], is on the

faculty at Abilene Christian University, Abilene, TX 79699. She is interested in preservice teacher preparation and teacher professional development. John Sampson, [email protected], teaches mathematics and is the

athletic director at Newellton High School, Newellton, LA 71357. He enjoys integrating new and exciting projects into his mathematics curriculum.

558 MATHEMATICS TEACHER



Year Amount_Description 0 A0 Deposit initial amount.

1 Ai=A0 + r ? A0 At the end of the first year, interest is calculated (r ? A0) and added to = A0(l + r) initial amount, resulting in the formula A0 + r ? A0, to obtain amount in

the account after one year, Aj. Simplify the expression by factoring.

2 A2=A + ?A At the end of two years, interest is calculated and added to present = Aid + r) amount in account to obtain new amount in account, (A + r? A ).

= A0( 1 + r)( 1 + r) Factor, then substitute A0( 1 + r) for Aj. = A0(l + r)2

3 A3=A2 + ?A2 Repeat. By factoring, then substituting A0(l + rf for A2, students can see = A2( 1 + r) that a pattern begins to emerge. = A0(l + r)2(l + r) = A0(l + r)3

An = A0(l + r)n Generalize pattern in recursive sequence to a closed form.

Fig. 1

Generalizing the sequence of the accumulated amount of interest

extend to modeling debt amortization that involves both geometric and arithmetic sequences. Students can then apply their knowledge of sequences, note

step-by-step change, and apply their observations to a relevant situation. Studying amortization of debt in this setting allows students to see that func tions can be defined implicitly, which may be a more natural way than their mathematically equiv alent explicit definitions. Finally, the amortization of credit-card debt integrates discrete mathematics into an algebra course, thereby allowing students to draw on various aspects of mathematics (NCTM 2000, pp. 31, 38, 70,289,305).

The teacher should begin the activity by initiat

ing a question-and-answer session to ascertain stu dents' knowledge concerning credit cards. Part of this discussion might address issues that affect the

economy, such as consumerism, credit-card debt, and loan-payment evasion. The teacher should give groups of four students a laminated "credit card" with a credit limit of $4000 and an annual interest rate of 18.99 percent. We chose this particular cred it limit because a trivia question on a local radio station stated that the average credit-card debt per cardholder was $4000. In addition, such shopping facilities as the Internet or mail-order catalogs need to be available to students. Students are told to "shop" for whatever they want. After students

complete their shopping, they record their purchas es and calculate taxes, as well as such other

charges as shipping. In the next phase of the activity, the teacher asks

students what would happen if, for some reason, they could no longer use their credit cards. Reasons

might include the account's being closed because of

a stolen card, a parent's confiscating the card, or the student's reaching the credit limit. The credit limit is relevant since groups tend to "max out" their credit cards by charging merchandise costing $4000 although they are not told to do so. The dis cussion needs to guide students to realize that they must pay for future purchases in cash and that credit-card companies expect monthly payments to continue until the debt is paid. The teacher should tell students to find a minimum monthly payment that is appropriate for their income level and to

explain, using various aspects of mathematics, why they selected this payment as a minimum. The fol

lowing prompts assist students in setting up their

investigation. The answers given assume a mini mum monthly payment of $50 and an initial balance of $4000.

Write a recursive formula to represent the

unpaid monthly balances, aw if no monthly pay ment is made.

______ . 0>n = + 1 12 1 an-l

0l1899\

a0 = 4000

Choose a monthly payment amount. Write a recursive formula that represents the unpaid monthly balances, bn, if no interest is charged.

&n = ?Vi-50

?o = 4000

Write a recursive formula that models the

sequence of unpaid monthly balances, cn, if inter

Credit-card

companies

expect

monthly payments to

continue

until the debt is paid

Vol. 94, No. 7 ? October 2001 559



How long will you need to

pay off your credit-card

debt if you pay the

minimum

monthly payment?

est is charged and if the chosen payment is made.

, 0.1899\ r c? = ( 1 + *cn_i- 50 12 ; c0 = 4000

Using your calculator's syntax, rewrite the

unpaid monthly balance sequences, renaming an,

bn, and cn as u(n), vin), and win), respectively. Enter each of these sequences into your calcula tor. Figure 2 shows commands for the TI-83

graphing calculator.

8 E1 Sci En9 E 123456789

S^KIMjf Degree_ Fune Par Pol SEE. Connected IflBH !llEJH*fcll Cs?nul J] a+bi reA0i il Horiz 6-T

Pl?tl Plot2 Plot3 ?Min=0

0.1899/12) u<?Min>B<:4000> . <?) <?-1>-50 w<ftMin)B<:4000> w<n>=_

, Pioti M?? Mot3 0.1899/12) u<?Min)B<:4000> . < ) < -1>-50 v<nMin>B<4000> .

, .w<n)Bw<n-l>*<l+ 0.1899/12)-50

<? ) <4 >

(a) Sequence mode for recursion; dot mode for discrete graphing

(b) Function notation used for sequences an = u{n); a0 = u(nhA\n) = 4000

bn = v{n)\ b0 = v(nM\n) = 4000

cn = w(n); c0 = w{nM\n) = 4000

Fig. 2 TI-83 commands

Compare the symbolic, graphical, and tabular

representations of these sequences. How does the symbolic representation in each formula pertain to the definition of the unpaid monthly balance?

What type of sequences are u(n) and ( )

Why? What relationship, if any, does w(n) have to

u(n) and ( ) Note the sequential changes in the unpaid monthly balances for u(n), v(n), and w{n). Which affects u(n), v(n), and w(n) more?the constant ratio or the constant difference? Why?

How long will you need to pay off your credit card debt if you pay the chosen minimum month

ly payment? What determines the payoff time? Be able to justify your answer.

What minimum monthly payments might be

acceptable to you and to the credit card compa

ny? Why? To see multiple options for paying off credit-card

debt, groups give oral presentations in which they

explain their findings to the class. If students do not mention situations similar to those depicted in

figures 3 and 4, teachers should demonstrate sce

narios involving monthly payments of $50.00, $63.30, and $75.00 during closure of the activity. Graphical and tabular representations for such minimum monthly payments as $50.00, $63.30, and

$75.00 for the first forty-eight months assist stu dents in making connections between mathematical

u(n), unpaid monthly balances if no monthly payment v(n), unpaid monthly balances if no interest

w{n), unpaid monthly balances with interest and payment Window and Table settings for minimum monthly payments of $50.00, $63.30, and $75.00 [window wMin=0 nMax=48 PlotStart=l PlotStep=l Xnin=0

, Xnax=48 ?Xsg1=1_

WINDOW |tPlotStep=l Xnin=0 Xnax=48 Xscl=l Vnin=0 Vnax=8000 VSC1=1000

TABLE SETUP TblStart=0 ?Tbl=l Indpnt: K?SS f?sk iDepend: EBEE Ask

$50.00 minimum monthly payment

1063.3 H127.fi 1192.5 1259.3 1326.7 1395.1

3B50 3B00 3750 3700

y? Iv<?)l u<?) 1000 3950 3900 3B50 3B00 3750 3700

1013.3 1026.B 1010.5 1051.5 106B.fi 10B3


1063.3 1127.fi 1192.9 1259.3 1326.7 1395.1

1000 3936.7 3B73.1 3B10.1 3716.B 36B3.5 3620.2

3936.7 3B73.1 3B10.1 3716.B 36B3.5 3620.2


1063.3 1127.6 1192.9 1259.3 1326.7 1395.1

3925 3B50 3775 3700 3625 3550

3925 3B50 3775 3700 3625 3550

3976.1 3961.3 3952.1 3939.6 3927

Fig. 3

Representations of unpaid monthly balance for first forty-eight months

u(n), unpaid monthly balances if no monthly payment v(n), unpaid monthly balances if no interest

w(n), unpaid monthly balances with interest and payment Window and Table settings for minimum monthly payments of $50.00, $63.30, and $75.00

(WINDOW wMin=0 ?Max=120 PlotStart=l PlotStep=l Xnin=0

, Xnax=120 ?Xscl=10

IWINDOW tPlotStep=l Xnin=0 Xnax=120 Xscl=10 Vnin=0 Vnax=8000 Vscl=1008

ITABLE SETUP TblStart=114

, *Tbl=l Indpnt: gm? Ask Depend: limas Ask


Fig. 4

Representations for unpaid monthly balances over time

concepts and personal finance. The following list serves as a guideline for facilitating closure:

Connect ideas of common ratio, interest rate, and

geometric, or exponential, growth. Interest rate in this context is the common

ratio of the geometric sequence.




A common ratio with an absolute value greater than 1 yields increasing growth. In this example, the interest rate of

, 0.1899\ 12 /

increases the amount of debt exponentially as

each new month's balance is calculated

through repeatedly multiplying the rate by the previous month's balance.

The geometric sequence u(n) is represented

graphically and numerically in figures 3 and 4.

Connect ideas of common difference, minimum

monthly payment, and arithmetic, or constant,

growth. The minimum monthly payment in the context

of the problem is the common difference of the

arithmetic sequence. The common difference is less than 0, so con

stant growth is negative. In this example, the minimum monthly pay

ment reduces the amount of debt linearly.

Graphical and numerical representations of arithmetic sequences, ( \ with common dif ferences of-50, -63.30, and -75 are displayed in figures 3 and 4.

Note that features of win) relate to both a com mon ratio and a common difference.

The sequence of unpaid monthly balances, win), is the sum of a geometric sequence and an arithmetic sequence. However, it is not the sum of u(n) and v(n).

m w(n)_^i

+ ^^yw(n-i).5o

= ,( -1) + (04^ ?^( -1)-50

0.1899\ 12 j

i2 ; win - 1) + [w(n -1) - 50]

0.1899\

is a geometric sequence, whereas

win -1) - 50

is an arithmetic sequence. The sums of a geometric sequence and arith metic sequences with common differences of

-50, -63.30, and -75 are displayed in the

graphical and tabular representations of win) in figures 3 and 4.

Note the effect of geometric and arithmetic fea tures on the growth of win), that is, the differ ence between the amount of interest for one month and the minimum monthly payment caus es win) to increase, remain constant, or decrease.

One month's interest is calculated by

0.1899\ 12 )

wiO).

If the minimum monthly payment is less than

/0.1899\ i2 ;

' wi0\

then win) increases; if the minimum monthly payment is equal to

/0.1899\ 12-j

wi0\

then win) is constant; and if the minimum

monthly payment is greater than

/0.1899\ HT wi0\

then win) decreases.

Note acceptable minimum monthly payments. The minimum monthly payment must be greater than the first month's interest and less than or

equal to the initial amount of debt. In the context

of the problem illustration, 63.30 < minimum

monthly payment ̂ 4000.

Note that the length of time to pay off the debt is determined by the interest rate and the initial debt.

The growth of win) results from the contribu tion of the geometric feature,

0.1899\ 12 I

win -1),

and the arithmetic feature, the minimum

monthly payment. 1 If debt is to be amortized, the minimum

monthly payment must be greater than the first month's interest so that all calculations of

monthly interest are less than the minimum

monthly payment and win) decreases to 0. 1 The first month's interest is established by the interest rate and the initial amount owed,

0.1899 12 wi0\

that is, the geometric feature and initial condition.

Consequently, because of their influence on the minimum monthly payment, the interest rate and the initial debt determine the amortiza tion period.

Note students' reflections on the length of time needed to pay off credit-card debt.

Time works for the investor but works against the debtor.

A consumer should seek lower interest rates and should keep credit-card debt to a manage able level.

In analyzing change and the effect of the geomet ric and arithmetic components on the unpaid

Time works

for the investor

but works

against the debtor

Vol. 94, No. 7? October 2001 561



The rate of \ decline is

much lower at the

beginning of the payoff | period than

it is at the end

monthly balances, w(n), we suggest using the Time Value of Money functions, TVM Solver, on the TI-83. Commands for the TVM Solver involving a mini mum monthly payment of $75 are given in figure 5. Amortizing debt requires a future value of 0, and the internal workings of the calculator necessitate

representing one of the cash outflows as a negative value (Texas Instruments 1996). Interest, the geo metric feature, determines the amount of increase in each month's unpaid balance. Likewise, the min imum monthly payment, the arithmetic component, determines the amount of decrease in each month's

unpaid balance. Therefore, the sequential change, a

negative amount, of the unpaid monthly balance, w(n) - w(n - 1), is the opposite of the difference between the minimum monthly payment and

monthly interest,

minimum monthly payment - 0.1899\

12 I w(n -1).

This difference is called the principal payment. By looking at tables of monthly interest and principal payments, students can analyze how the unpaid monthly balances are changing at the beginning and end of the amortization period and why this

change occurs. See figure 6 for commands and tables. The calculator must be in function mode.

= number of payments. I = annual rate as a percent.

PV = present amount of debt as a cash outflow. PMT = monthly payment. FV = future amount of debt. P/Y = payments per year. C/Y = compounding periods per year. PMT: END BEGIN = payment made at end of period or

beginning of period.

Solve for N: Alpha-Enter I" N=l 18.3293163 / =18.99 Py= -4 0 PMT=75 FV=0 P/Y=12 C/V=12 _ PMT:WI< ?

Number of payments to amortize $4000 debt with a $75 minimum monthly payment

Fig. 5 -83's TVM-Solver?amortization of debt

In addition, the graph in figure 3 of the unpaid monthly balances with a $75 minimum monthly payment helps students connect the principal pay ment with the rate of decline of w(n). From the

tables, students can see that the interest is greater than the principal at the beginning of the amortiza tion period; but at the end of the period, the princi pal is greater than the interest. Therefore, the rate of decline is much lower at the beginning of the

payoff period than it is at the end. The graph depicts this rate of decline by its "flatness" near the

beginning of amortization in comparison with its

steepness near the month in which payoff occurs.

Vagi URRS 6Ttvn_FV 7:npv< 8:irr< 9:bali

niPrn< ,.. ZInU | 4> <

Finance Menu Pioti Mot2 Mot3

, ) NVzBZPrnCX,*) SV3B75 nVh = .Vs = V6 = W=_

5.00 6.00 ?.00

Vi 63.30 63.11 62.93 6271 62.51 62.31 62.11

V2 11.70 11. b9 12.0? 12.26 12.16 12.66 12. b6

(X,X): interest added each month to unpaid monthly balance, that is, amount contributed by common ratio, geo metric component IPRN(X.X): amount of payment applied to unpaid monthly balance, that is, the absolute value of the rate of change of

unpaid monthly balances, or the difference between the minimum monthly payment and monthly interest $75: minimum monthly payment, arithmetic component of w{n)

Fig. 6

Analyzing change using the TVM Solver

Also, the rate of decline over time for w(n) is not

constant, so the constant difference, the arithmetic characteristic of the sequence, appears to be over

shadowed by some other feature. The graph of w(n)

supports this observation, since it does not exhibit the behavior of an arithmetic sequence, such as

( ). An analysis of the rate of decline by examining the calculation of the minimum monthly payment in terms of its two components, the monthly inter est and the principal payment, reveals what is

occurring. We let prn(n) represent the principal payment for month n. For the first month, the mini mum monthly payment is

0.1899\ i2 y

w(0) + prnil).

Reasoning through the makeup of the minimum

monthly payment for the second month reveals that

part of the first month's interest becomes part of the principal payment in subsequent months. Also, the principal payment, prn(n), is the opposite of

wWs arithmetic change, win) - w(n -1), that is, prn(n) = [w(n -1)

- w(n)]. Therefore, the second

month's minimum monthly payment is calculated as follows:

0.1899\ i2 y

w(l) + prn(2)

01899\ ? w(l) + (minimum monthly payment ' - interest for month 2)

12

0.1899\ 12

' w(l) + 0.1899 12

w(0) + prnd)

0.1899\, 12 /

i?(l)

= [^yw(iu[^)[w(0)-w(l)]+prn(l)




Therefore, the amount applied toward the principal in the second month is

(lto^?).,,?<?. Similarly, the minimum monthly payment for

the third month is calculated as follows:

p^)-u;(2)+pm(3) (0 1899\ ? ? ; (2) + (minimum monthly payment - interest for month 3)

(0^1). ?(2)^0^. ?(0)^1)

-|^f).,(2)]

Jo^L? jo^ . 0)-?a)+?a) \ U / \ U I -w(2)]+prn(l)

+ (^^)-MD-^(2)]+pm(l)

pm(2)+pm(l)

= (2^!).?(2)

+ (u?f?).pma)*(2f?)

prn(2)

--[^yw(2)+Prn(2)[^yPrn(2)

Continuing, the minimum monthly payment for ich month is calculated by

^onunuir

each month

/0.1899\ , 0.1899\ (

10.1899/12 of the previous i to interest and

-1),

month's which means that 0.1899/12 of balance is applied to interest and

of the previous month's principal payment is

applied to the balance of the debt. So the sequence of principal payments is a geometric sequence with the same common ratio as the geometric component of w(n); and with a minimum monthly payment of

$75, it has an initial condition of pra(l) = $11.70. With the TI-83, students can check the common ratio by using the XPRN( command, as shown in

figure 7.

Moti Pioti Plot3 nV2 = \Y3BZPrn<X>X>

Wi= sVsBZPrn(X+l,X+l

Wfi = .V? =

12.073 12.261 12.15B 12.656 12.B56 |Vs=l. 815825

Fig. 7

Finding the common ratio for principal payment

By analyzing the calculation of w(n) in general terms of monthly interest and principal payment, we can show that the geometric features of w(n) do overshadow its arithmetic feature. Over time, larg er amounts of the minimum monthly payment, the arithmetic feature, are applied to the principal pay ment, which is geometric in nature. The following is a calculation for the unpaid monthly balance for

month n:

w(n) 01899\ - minimum monthly 12 J payment

i2 ; - (monthly interest + principal)

0.1899. = 1^-1) +M? )-w(n-l) 12 w(n-l)

+|1+o^j+pm&l_1) / ,x /0.1899\ / ix /0.1899 = w(n-l) +

h^j^ rw(n-D-?j7p

w (n -1) -11

+ 0 J^j-

+ prn(n -1)

= w(n-l)-[u0-^yPrri(n-l)

For homework, the teacher can give students

copies of various credit-card offers that show the interest rate and credit limit. Offers for entering college freshmen are preferable. Students write a

paragraph explaining the amount of debt that they incurred on the account, their choice of a minimum

monthly payment, the length of time for payoff, and the mathematical reasons for their answers. Also, they should include a reflection on what they learned from the activity.

Credit-card debt transcends all geographic regions and economic levels, making it a topic wor

thy of discussion in mathematics classes in grades 9-12. Through this activity, second-year algebra students can investigate credit-card debt as an

Credit-card

debt transcends

all

geographic regions and

economic

levels

Vol. 94, No. 7? October 2001 563



Write. Do Math. Communicate. Create.

Scientific Notebook software enables you to: Write text and math easily

Do math with the easy-to-use computer algebra system Publish math on the web

Create exams

www.mackichan.com/mt Email: [email protected] Toll Free: 877-724-6683 Fax: 206-780-2857

application of geometric and arithmetic sequences. By doing so, secondary mathematics can educate our younger citizens about a relevant and important economic issue.

REFERENCES Consumer Federation of America. "Credit Card Debts Escalate

in 1997." 16 December 1997. www.consumerfederation.org /backpage/press.html. World Wide Web.

-. "Credit Card Debt Imposes Huge Costs on Many College Students: Previous Research Understates Extent of Debt and Related Problems.* 8 June 1999. www

.consumerfederation.org/backpage/press.html. World Wide Web.

Feldstein, Stuart A "The Rise in Personal Bankruptcy: Causes and Impact." Report to Subcommittee on Commercial and Administrative Law, U.S. House of Representatives. 10 March 1998. www.house.gov/judiciary/5178.htm. World Wide Web.

Johnson, David B., and Thomas A. Mowry. Mathematics: A Practical Odyssey. 3rd ed. Pacific Grove, Calif.: Brooks/Cole Publishing Co., 1998.

National Council of Teachers of Mathematics (NCTM). Cur riculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989.

-. Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

Nellie Mae. "Study Shows Student Credit Card Debt Rising." Press release dated 13 February 2001. Available at

www.prnewswire.com. World Wide Web. Texas Instruments. TI-83 Graphing Calculator Guidebook. N.P.: Texas Instruments, 1996. ?

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